$|G_2:G_3|$ can be an arbitrarily large finite number (although I don't know exactly what values it could take). If there is a finitely presented infinite simple group generated by elements of order $3$, then it can also be infinite.
I found an example of a group $P$ of order $3^4$ satisfying your hypotheses, but that has $|P_2:P_3|=3$,
If we take a wreath product $W$ of a simple group $S$ that can be generated by elements of order $3$ with $P$ in a degree $27$ permuation representation, then $|W_0:W_1|=|W_1:W_2|=3$, but $|W_2:W_3| = 3|S|^3$. This is verified for $S=A_5$ in the Magma calculation that follows.
> F:=Group<x,y|x^3,y^3>;
> P:=pQuotient(F,3,3);
> P:=quo<P|P.5>;
> #P;
81
> Q:=CosetImage(P,sub<P|P.1>);
> W:=WreathProduct(Alt(5),Q);
> [Order(W.i): i in [1..Ngens(W)]];
[ 3, 3, 3, 3, 3, 3 ]
> N:=ncl<W|W.1>;
> Index(W,N);
3
> N2:=ncl<N|W.1>;
> Index(N,N2);
3
> N3:=ncl<N2|W.1>;
> Index(N2,N3);
648000